2XV International Congress on Mathematical Physics Plenary Lectures

2.0.1Mathematical Developments Around the Ginzburg-Landau Model in 3D Program

Jean Bourgain

IAS, Princeton bourgain@ias.edu

We are discussing the 3D Ginzburg Landau functional without magnetic field in the London limit,which is the 3D counterpart of the work of Bethuel-Brezis-Helein.In particular the role of the minimal connection in the evaluation of the GinzburgLandau energy is explained and optimal regularity properties of the minimizers stated. A key role is played by certain novel Hodge decompositions at the critical Sobolev index.

2.0.2 The Riemann-Hilbert Problem: Applications

Percy Deift

Courant Institute, NYU deift@courant.nyu.edu

In this talk the speaker will describe the application of Riemann-Hilbert techniques to a variety of problems in mathematics and mathematical physics. Algebraic and analytical applications will be discussed. The nonlinear steepest descent method plays a key role.

2.0.3 Fluctuations and Large Deviations in Non-equilibrium Systems

Bernard Derrida

ENS, Paris derrida@lps.ens.fr

Systems in contact with two heat baths at unequal temperatures or two reservoirs of particles at unequal densities reach non-equilibrium steady states. For some simple models, one can calculate exactly the large deviation functions of the density profiles or of the current in such steady states.

These simple examples show that non-equilibrium systems have a number of properties which contrast with equilibrium systems: phase transitions in one dimension, non local free energy functional, violation of the Einstein relation between the compressibility and the density fluctuation, non-Gaussian density fluctuations.

In collaboration with T. Bodineau, J.L. Lebowitz, E.R. Speer.

2.0.4Hamiltonian Perturbations of Hyperbolic PDE’s: from Classification of Equations to Properties of Solutions

Boris Dubrovin

SISSA, Trieste dubrovin@sissa.it

The talk will deal with the classification of Hamiltonian perturbations of hyperbolic PDEs with one spatial dimension and with the comparative study of their singularities.

2.0.5Spontaneous Replica Symmetry Breaking in the Mean Field Spin Glass Model

Francesco Guerra

Universita Roma 1 “La Sapienza”

Francesco.Guerra@roma1.infn.it

We give a complete review about recent methods and results in the mean field spin glass theory. In particular, we show how it has been possible to establish the rigorous validity of the Parisi representation for the free energy in the infinite volume limit. The origin of the Parisi functional order parameter is explained in the frame of Derrida-Ruelle probability cascades. Finally, we conclude with an outlook on possible future developments.

2.0.6Spectral Properties of Quasi-Periodic Schrödinger Operators: Treating Small Denominators without KAM

Svetlana Jitomirskaya

UC-Irvine szhitomi@math.uci.edu

Two classical small divisor problems arise in the study of spectral properties of quasiperiodic Schroedinger operators, one related to Floquet reducibility (for low couplings of the potential term), and the other related to localization (for high couplings). Both have been traditionally attacked by sophisticated KAM-type methods.

In this talk I will discuss more recent non-KAM based methods for both localization and reducibility, that are significantly simpler and lead, where applicable, to stronger results. In particular they usually lead to so-called nonperturbative (i.e. uniform in the Diophantine frequency) estimates on the coupling, and sometimes to the results covering the entire expected region of couplings.

I will discuss the recent joint work with A. Avila on nonperturbative reducibility, with various sharp spectral consequences in the low coupling regime, and review earlier results by J. Bourgain, M. Goldstein, W. Schlag, and the speaker, on nonperturbative localization.

2.0.7 Conformal Field Theory and Operator Algebras

Yasuyuki Kawahigashi

University of Tokyo yasuyuki@ms.u-tokyo.ac.jp

Algebraic quantum field theory is an operator algebraic approach to quantum field theory and its main object is a family of operator algebras parameterized by spacetime regions, rather than Wightman fields. I will present recent progress on classification of conformal field theories within this approach.

Chiral, full and boundary conformal field theories are described in a unified framework and we present their complete classification for the case where the central charge is less than 1. In the chiral case, our classification list contains an example which does not seem to be known in other approaches to conformal field theory. Our tools are based on operator algebraic representation theory and are applications of the Jones theory of subfactors. Similar methods are also useful for operator algebraic studies of the Moonshine conjecture.

This is a joint work with Roberto Longo.

2.0.8Random Schrödinger Operators, Localization and Delocalization, and all that

Abel Klein

UC, Irvine aklein@math.uci.edu

In the widely accepted picture of the spectrum of a random Schrödinger operator in three or more dimensions, there is a transition from an insulator region, characterized by “localized states”, to a very different metallic region, characterized by “extended states”. The energy at which this metal-insulator transition occurs is called the mobility edge. The standard mathematical interpretation of this picture translates “localized states” as pure point spectrum with exponentially decaying eigenstates (Anderson localization) and “extended states” as absolutely continuous spectrum.

Forty some years have passed since Anderson’s seminal article but our mathematical understanding of this picture is still unsatisfactory and one-sided: the occurrence of Anderson localization is by now well established, but with the exception of the special case of the Anderson model on the Bethe lattice, there are no mathematical results on the existence of continuous spectrum and a metal-insulator transition.

In this lecture I will first review localization, including the recent proofs of localization for the Bernoulli-Anderson Hamiltonian (Bourgain and Kenig) and for the Poisson Hamiltonian (Germinet, Hislop and Klein).

I will then present an approach to the metal-insulator transition based on dynamical (i.e., transport) instead of spectral properties (Germinet and Klein). Here transport refers to the rate of growth, with respect to time, of moments of a wave packet initially localized both in space and energy. The region of dynamical localization is

defined to be the spectral region where the random Schrödinger operator exhibits strong dynamical localization, and hence no transport. The region of dynamical delocalization is the spectral region with nontrivial transport. There is a natural definition of a dynamical metal-insulator transition and of a dynamical mobility edge. We proved a structural result on the dynamics of Anderson-type random operators: at a given energy there is either dynamical localization or dynamical delocalization with a nonzero minimal rate of transport. The region of dynamical localization turns out to be the analogue for random operators of Dobrushin-Shlosman’s region of complete analyticity for classical spin systems, and may be called the region of complete localization.

I will close with the proof of the occurrence of this dynamical metal-insulator transition in random Landau Hamiltonians (Germinet, Schenker and Klein). More precisely, we show the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level, which combined with the known dynamical localization at the edges of each disordered-broadened Landau band, implies the existence of at least one dynamical mobility edge in each Landau band.

2.0.9 Perelman’s Work on the Geometrization Conjecture

Bruce Kleiner

Yale bruce.kleiner@yale.edu

The lecture will discuss the Ricci flow approach to Geometrization.

2.0.10 Trying to Characterize Robust and Generic Dynamics

Enrique Ramiro Pujals

IMPA, Rio de Janeiro enrique@impa.br

If we consider that the mathematical formulation of natural phenomena always involves simplifications of the physical laws, real significance of a model may be accorded only to those properties that are robust under perturbations. In loose terms, robustness means that some main features of a dynamical system (an attractor, a given geometric configuration, or some form of recurrence, to name a few possibilities) are shared by all nearby systems.

In the talk, we will explain the structure related to the presence of robust phenomena and the universal mechanisms that lead to lack of robustness. Providing a conceptual framework, the goal is to show how this approach helps to describe “generic” dynamics in the space of all dynamical systems.